WORST_CASE(?,O(n^1))
* Step 1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(+) = [1] x1 + [0]
               p(0) = [0]         
               p(s) = [1] x1 + [4]
             p(sum) = [4] x1 + [0]
            p(sum1) = [4] x1 + [1]
          
          Following rules are strictly oriented:
           sum(s(x)) = [4] x + [16]        
                     > [4] x + [0]         
                     = +(sum(x),s(x))      
          
           sum1(0()) = [1]                 
                     > [0]                 
                     = 0()                 
          
          sum1(s(x)) = [4] x + [17]        
                     > [4] x + [5]         
                     = s(+(sum1(x),+(x,x)))
          
          
          Following rules are (at-least) weakly oriented:
          sum(0()) =  [0]
                   >= [0]
                   =  0()
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(0()) -> 0()
        - Weak TRS:
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(+) = [1] x1 + [0]
               p(0) = [0]         
               p(s) = [1] x1 + [0]
             p(sum) = [11]        
            p(sum1) = [0]         
          
          Following rules are strictly oriented:
          sum(0()) = [11]
                   > [0] 
                   = 0() 
          
          
          Following rules are (at-least) weakly oriented:
           sum(s(x)) =  [11]                
                     >= [11]                
                     =  +(sum(x),s(x))      
          
           sum1(0()) =  [0]                 
                     >= [0]                 
                     =  0()                 
          
          sum1(s(x)) =  [0]                 
                     >= [0]                 
                     =  s(+(sum1(x),+(x,x)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))